The non-destructive investigation of samples is an important object in various technical fields like material sciences, medical examinations, archaeology, construction technique, techniques concerning security matters etc. One approach for obtaining an image of a sample e.g. by computer tomography (CT) is based on an irradiation trough a sample plane from different projection directions with X-rays, followed by the reconstruction of the sample plane on the basis of attenuation data measured at different directions. The entirety of the measured attenuation data can be described in terms of so-called Radon data in a Radon space.
Different reconstruction methods for Radon data are known today. For an introduction to the mathematical and physical principles of conventional image reconstruction, reference is made to the textbooks “Computed Tomography—Fundamentals, System Technology, Image Quality, Applications” by W. A. Kalender (1st. edition, ISBN 3-89578-081-2); “Image Reconstruction from Projections: The Fundamentals of Computerized Tomography” by G. T. Herman, Academic Press, 1980; and “Einführung in die Computertomographie” by Thorsten M. Buzug (Springer-Verlag, Berlin 2004). The conventional reconstruction methods can be summarized as methods based on the iterative reconstruction or those based on the so-called filtered back-projection.
The iterative reconstruction is an approximation method based on a plurality of iteration steps. Each point in a projection corresponds to a line in the reconstructed image. The projections are thus back-projected. This leads as a first step to a very crude approximation. Subsequently, the imaging process of transforming the Radon data is simulated for this approximation and then differences are calculated to do a back-projection again. For an optimization of the reconstructed picture, this iteration is repeated many times. The essential disadvantage of the iterative reconstruction is that the above iteration leads to extremely long calculation times.
The filtered back-projection method relies in principle on the Fourier-slice theorem describing a relationship of the Fourier transform of the Radon data and Fourier transformed image data. A general disadvantage of using the Fourier-slice theorem lies in the fact that an interpolation step in the reconstruction results in errors and artifacts which have a tendency even to increase with increasing space frequency. The capability of reconstructing images with fine details is limited. This disadvantage could be avoided by using detectors with high resolution only. However, the application of these detectors is limited in terms of dose burden, costs and data processing time. Another problem is related to the discretization of the Radon data from which the image data have to be reconstructed. To get an optimal filtered back-projection reconstruction it would be necessary to exactly match the projected irradiation rays with the sensor elements of the detectors. This is in general not the case. For this reason, uncertainties or smoothing effects from the reconstruction of Radon data by means of filtered back-projection algorithms are introduced. This drawback can in general not be overcome by filtered back-projection algorithms. It could be avoided by using the above mentioned iterative reconstruction but these are so computational expensive that they are not widely used in practical computed tomography.
The so-called Feldkamp algorithm or the advanced single slice reconstruction are methods that try to adapt the filtered back-projection algorithm to the case where the data are collected in helical computed tomography with fan or cone beam geometry which results in data points not evenly spread within the z-axis direction and the rays along which the projection and integrations take place are tilted against each other. According to the Fourier-slice theorem all possible rays have to be evaluated, because otherwise the error for high space frequencies would be larger. This leads to further uncertainties and unsharpness.
Generally, the conventional techniques allow that the unsharpness of the structure reconstruction can be reduced, but not avoided, by using algorithms with a higher need of computational power.
Current developments in computed tomography have provided so-called multi-slice-CT and CT-systems based on flat panel technology. These developments are suffering from two further major problems. First of all, the amount of data is very large, the reconstruction time for such an amount of data is too long or the computers needed to handle such data are too expensive. The second problem is scatter radiation. Scatter radiation becomes a bigger problem for larger radiated areas of the object. Conventional scatter reduction methods like grids do not have enough effect and the signal-to-noise ratios are anyhow already very small, which means that a further reduction of signal as resulting from inserting a grid would produce more artifacts and worse images. Additionally, assuming an oscillation of the grid relative to the detector to avoid grid lines, the rotation of the grid with the detector is very difficult in consideration of the rotation time of 0.5 s or less.
The above disadvantages are associated not only with the conventional CT imaging, but also with all available reconstruction methods related to Radon data. As an example, a practical reconstruction technique for neutron transmission imaging is not available at present. As a further example, the application of conventional ultrasound tomography e.g. in medicine is restricted to small or soft tissue objects since, the conventional reconstruction techniques require integrated projection data from the centre of the object to be investigated.